2015/02/02 by Andrew Lawrie, Lawrie, Andrew, Sung‐Jin Oh +4
Mathematics · Physics and Astronomy · #47F05 #58J45 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.CA #math.SP #msc:47F05 #msc:58J45
paper · pdf · doi:10.48550/arxiv.1502.00697
arXiv admin note: text overlap with arXiv:1402.5981
openalex publication_date 2015/02/02 · arxiv created 2015/02/03 · arxiv updated 2015/02/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study k-equivariant wave maps from the hyperbolic plane into the 2-sphere as well as the energy critical equivariant SU(2) Yang-Mills problem on 4-dimensional hyperbolic space. The latter problem bears many similarities to a 2-equivariant wave map into a surface of revolution. As in the case of 1-equivariant wave maps considered in~\citeLOS1, both problems admit a family of stationary solutions indexed by a parameter that determines how far the image of the map wraps around the target manifold. Here we show that if the image of a stationary solution is contained in a geodesically convex subset of the target, then it is asymptotically stable in the energy space. However, for a stationary solution that covers a large enough portion of the target, we prove that the Schrödinger operator obtained by linearizing about such a harmonic map admits a simple positive eigenvalue in the spectral gap. As there is no a priori nonlinear obstruction to asymptotic stability, this gives evidence for the existence of metastable states (i.e., solutions with anomalously slow decay rates) in these simple geometric models.